Differential equations by direct integration
Problem 6.684 · easy
Solve \( \displaystyle y' = - 2 x + \cos{\left(x \right)} \) with \( \displaystyle y(0) = -4 \).
- \[ \frac{d}{d x} \left(- x^{2} + \sin{\left(x \right)}\right) = - 2 x + \cos{\left(x \right)} \]Integrate: y = this + C.✓ Proved
- \[ - \left. - x^{2} + \sin{\left(x \right)} \right|_{\substack{ x=0 }} - 4 = -4 \]y(0) = -4 fixes C.✓ Proved
- \[ \left. - x^{2} + \sin{\left(x \right)} - 4 \right|_{\substack{ x=0 }} = -4 \]The initial condition holds.✓ Proved
Answer \( y = - x^{2} + \sin{\left(x \right)} - 4 \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | finite differences of the answer match f, and the initial value holds |
Reviewers
gpt-oss:20b: fail (misleading) — The solution skips the step of introducing the constant of integration C and incorrectly uses a subtraction of 4 to satisfy the initial condition. While the final result is correct, the reasoning does not properly justify how C is determined, potentially misleading a student.qwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative and verifies the initial condition. The logic is sound and the steps are valid.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the antiderivative and verifies the initial condition. The logic is sound and the steps are valid.gpt-oss:20b: fail (misleading) 2026-10-19 — The solution skips the step of introducing the constant of integration C and incorrectly uses a subtraction of 4 to satisfy the initial condition. While the final result is correct, the reasoning does not properly justify how C is determined, potentially misleading a student.qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the general antiderivative and verifies that the specific constant satisfies the initial condition. The logic is sound and the steps are valid.gpt-oss:20b: fail (error) 2026-10-17 — The solution omits the constant of integration C in the general antiderivative. It incorrectly deduces C by subtracting 4 from the evaluated antiderivative at 0, rather than setting y(0)=C and solving C=-4. This misrepresents the derivation of the particular solution.qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the antiderivative and verifies the initial condition. The comments accurately describe the steps taken.gpt-oss:20b: pass 2026-10-17
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/de_direct_integration, checked 2026-10-19 with SymPy 1.14.0.